Download WORK/ENERGY THEOREM

WORK/ENERGY THEOREM There are a number of ways to determine the net work done on a system. One of the many would be to sum the forces vectorially, then dot that net force into the displacement. That is: !
!
Wnet = Fnet • d
To make the calculaFons easier, I’m going to assume the displacement and the net force are in the same direcFon (this will make the angle between the two zero). This lets us do the dot product: !
!
Wnet = Fnet • d
!
!
= Fnet d
( cos 0 )
o
= ma d
1.) Assuming the net force is constant, the acceleraFon will be constant. That means we can use kinemaFcs. Remembering that: a=
v2 − v1
Δt
⎡ ( v2 + v1 ) ⎤
d = vavg Δt = ⎢
⎥ Δt
2
⎣
⎦
and From that, the net work can be wriPen: Wnet = ma d
⎛ v2 − v1 ⎞ ⎛ ⎛ v2 + v1 ⎞ ⎞
= m⎜
Δt
⎝ Δt ⎟⎠ ⎜⎝ ⎜⎝ 2 ⎟⎠ ⎟⎠
(
)
1
m v22 − v12
2
1
1
= mv22 − mv12
2
2
=
2.) This expression, 1 2 1 2
Wnet = mv2 − mv1
2
2
Is called the work/energy theorem. 1 2
The mv
term is important enough to be given name. It is called kine/c energy. 2
With it, we can write the work/energy theorem as: Wnet = KE 2 − KE1
= ΔKE
In short, the work/energy really states that when there is net work done on a body, its kineFc energy will change. 3.) For those who are interested in such things, the work/energy theorem is typically derived using Calculus. Specifically: !
!
Wnet = ∫ Fnet • dr
=
∫ ( ma ) dr
chain rule ⎛ dv ⎞ ⎛ dr ⎞
= m ∫ ⎜ ⎟ ⎜ dt ⎟
⎝ dt ⎠ ⎝ dt ⎠
⎛ dv ⎞
= m ∫ ⎜ ⎟ ( v dt )
⎝ dt ⎠
v2
= m ∫ v dv
v1
v2 v2
=m
v
2 1
1
1
= mv22 − mv12
2
2
4.)